2. Homotopy groups in pro-artinian categories

M IC H E L B R IO N

O N T H E F U N D A M E N TA L G R O U P S O F C O M M U TAT I V E A L G E B R A IC G R O U P S

S U R L E S G R O U P E S F O N D A M E N TA U X D E S G R O U P E S A L G É B R IQ U E S C O M M U TAT IF S

Abstract. — Consider the abelian category C of commutative group schemes of finite type over a field k, its full subcategory F of finite group schemes, and the associated pro- category Pro(C) (resp. Pro(F)) of pro-algebraic (resp. profinite) group schemes. When k is perfect, we show that the profinite fundamental group $₁ : Pro(C)→ Pro(F) is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors$_ivanish fori>2. Along the way, we describe the indecomposable projective objects of Pro(C) over an arbitrary fieldk.

Résumé. — Considérons la catégorie abélienneCdes schémas en groupes de type fini sur un corpsk, la sous-catégorie pleineF des schémas en groupes finis, et la catégorie correspondante Pro(C) (resp. Pro(F)) des groupes proalgébriques (resp. profinis). Lorsquek est parfait, nous montrons que le groupe fondamental profini$₁ : Pro(C) → Pro(F) est exact à gauche et commute aux extensions algébriques de corps ; il en résulte que les groupes d’homotopie profinis supérieurs$_i sont nuls pour i>2. Au passsage, nous décrivons les objects projectifs indécomposables de Pro(C) sur un corpskarbitraire.

Keywords:commutative algebraic groups, fundamental groups.

2010Mathematics Subject Classification:14K05, 14L15, 18E15, 20G07.

DOI:https://doi.org/10.5802/ahl.25

1. Introduction

Every real Lie groupG gives rise to two exact sequences

0→G⁰ →G→π₀(G)→0, 0→π₁(G)→G^e →G⁰ →0,

where G⁰ denotes the identity component, G^e its universal cover, and π₀(G), π₁(G) are discrete groups; moreover, the second homotopy group π₂(G) vanishes. This classical result has a remarkable analogue for commutative algebraic groups over an algebraically closed field k, as shown by Serre and Oort via a categorical ap- proach (see [Ser60, Oor66]). More specifically, consider the category C of commu- tative k-group schemes of finite type, and the full subcategory F of finite group schemes; then C is an artinian abelian category, and F is a Serre subcategory. Let Pro(C) (resp. Pro(F)) denote the associated pro-category, consisting of pro-algebraic (resp. profinite) group schemes; recall that these categories have enough projectives, andC (resp. F) is equivalent to the full subcategory of Pro(C) (resp. Pro(F)) con- sisting of artinian objects. Assigning to each object of Pro(C) its largest profinite quotient yields a right exact functor

$₀ : Pro(C)−→Pro(F).

It turns out that the left derived functors,

$_i :=Lⁱ$₀ : Pro(C)−→Pro(F),

vanish for i > 2; equivalently, $₁ is left exact. Moreover, $₀, $₁ fit in an exact sequence

0−→$₁(G)−→G^e −→G−→$₀(G)−→0

for any G∈Pro(C) (see [Ser60, 6.2, 10.2] when k has characteristic 0, and [Oor66, II.7, II.14] in positive characteristics).

The construction of the “profinite homotopy functors” $_i makes sense over an arbitrary fieldk; it is easy to extend the above exact sequence to this setting. The main result of this paper generalizes those of Serre and Oort as follows:

Theorem 1.1. — When k is perfect, the functor $₁ : Pro(C) → Pro(F) is left exact and commutes with base change under algebraic field extensions. As a consequence, the higher profinite homotopy functors $_i vanish for i>2.

Our approach is independent of the general theory of étale homotopy groups of schemes (see e.g. [AM69, Fri82]). We rather develop an ad hoc theory of homotopy groups in the setting of pairs (A,B), where A is an artinian abelian category, and B a Serre subcategory of A. For this, we build on constructions and results of Gabriel (see [Gab62, Chap. III]) and on further developments in [Bri19], recalled in Subsection 2.1. These may be conveniently formulated in terms of orthogonal or perpendicular categories (see [BR07, II.2] and [GL91] for these two notions).

Homotopy groups are introduced in Subsection 2.2, which generalizes results of Demazure and Gabriel on the profinite homotopy groups of affine group schemes (see [DG70, V.3.3]). Subsection 2.4 investigates compatibility properties of homotopy

groups in the presence of a Serre subcategoryC of B.

In Section 3, we first apply this formalism to the categoryC of (commutative) alge- braic groups, and its full subcategory L of linear algebraic groups, over an arbitrary fieldk; then Pro(L) is equivalent to the category of affine k-group schemes, in view of [DG70, V.2.2.2]. The resulting homotopy functor π₁^C,L turns out to be left exact (Proposition 3.3). We then consider the pair (C,F), and obtain the left exactness of

$₁ =π^C,F₁ when k is perfect; in addition, we show that the profinite universal cover Ge has homological dimension at most 1 for any G∈Pro(C) (Theorem 3.5).

When G is an abelian variety over an arbitrary field k, we construct a minimal projective resolution of G^e (Theorem 3.10). We also describe the projective objects of Pro(C) (Proposition 3.11); for this, we use results of Demazure and Gabriel on the projectives of Pro(L) over a perfect field (see [DG70, V.3.7]), combined with properties of the isogeny category C/F (see [Bri17]). We then show that the profinite homotopy functors commute with base change under separable algebraic field extensions (Proposition 3.15), thereby completing the proof of the main result.

As an application of the above developments, we obtain a spectral sequence à la Milne (see [Mil70]), which relates the extension groups inC and in the corresponding category over a Galois extension of k. Further applications, to the structure of homogeneous vector bundles over abelian varieties, are presented in [Bri18].

When the ground fieldk has characteristic p > 0, the prime-to-p part$^(p₁ ⁰⁾ of the profinite fundamental group commutes with arbitrary field extensions, and hence is left exact (Proposition 3.17). But over an imperfect fieldk, the functors $₀, $₁ do not commute with purely inseparable field extensions, nor does the pro-étale p-primary part of $1 (see Remarks 3.19, 3.20 and 3.21). In this setting, it seems very likely that $2 is nontrivial, but we have no explicit example for this; also, the profinite fundamental group scheme $₁ deserves further investigation, already for smooth connected unipotent groups.

Finally, it would be interesting to relate the above (affine, profinite or pro-étale) fundamental groups with further notions of fundamental group schemes considered in the literature. In this direction, note that the profinite fundamental group of any abelian variety A coincides with Nori’s fundamental group scheme (defined in [Nor76, Nor82]), as shown by Nori himself in [Nor83]. Also, whenkis algebraically closed, the affine fundamental group of A coincides with its S-fundamental group scheme introduced by Langer in [Lan11], as follows from [Lan12, Thm. 6.1].

Acknowledgments. Many thanks to Cyril Demarche, Mathieu Florence, Roy Joshua, Bruno Kahn, Chu Gia Vuong Nguyen and Takeshi Saito for very helpful discussions on the topics of this paper. Special thanks to the referee for a careful reading of the paper and valuable comments.

2. Homotopy groups in pro-artinian categories

2.1. Pro-artinian categories and colocalizing subcategories

Consider an artinian abelian category A, and the associated pro-category Pro(A).

Then Pro(A) is a pro-artinian category in the sense of [DG70, V.2.2]; equivalently,

the opposite category is a Grothendieck category. Moreover, A is equivalent to the Serre subcategory of Pro(A) consisting of artinian objects (see [DG70, V.2.3.1]). Let B be a Serre subcategory of A; then we may view Pro(B) as a Serre subcategory of Pro(A), stable under inverse limits (see [Bri19, Lem. 2.11]). We denote by ^⊥Pro(B) the full subcategory of Pro(A) with objects those X such that Hom_Pro(A)(X, Y) = 0 for all Y ∈ Pro(B) (this is the left orthogonal subcategory to Pro(B) in Pro(A) in the sense of [BR07, II.1]).

Lemma 2.1. — Let X ∈Pro(A).

(1) X ∈^⊥Pro(B) if and only if Hom_Pro(A)(X, Y) = 0 for all Y ∈ B.

(2) X has a smallest subobject X^B in Pro(A)such that X/X^B ∈Pro(B). More- over,X^B ∈^⊥Pro(B).

(3) For any morphism f :X →Y inPro(A), we havef(X^B)⊂Y^B with equality if f is an epimorphism. If in addition f is essential and Y ∈ ^⊥Pro(B), then X ∈^⊥Pro(B).

Proof. —

(1). — Let Y ∈ Pro(B). Then Y = lim←Y_i, where Y_i ∈ B. Therefore, we have HomPro(A)(X, Y) = lim←HomA(X, Y_i) = 0.

(2). — Let (X_i)i∈I be a family of subobjects of X such that X/X_i ∈Pro(B) for all i. ThenX/(∩i∈IX_i) is a subobject of ^Q_i∈IX/X_i, and hence an object of Pro(B).

This shows the existence ofX^B.

If there exists a nonzero morphism f : X^B → Y for some Y ∈ Pro(B), then X⁰ := Ker(f) is a subobject ofX^B such thatX^B/X⁰ is a nonzero object of Pro(B). It follows thatX/X⁰ ∈Pro(B), contradicting the minimality of X^B. So X^B ∈^⊥Pro(B).

(3). — The composition X^B → X → Y → Y /Y^B is zero, hence f(X^B) ⊂ Y^B. If f is an epimorphism, then it induces an epimorphism X/X^B → Y /f(X^B). So Y /f(X^B) ∈ Pro(B), i.e., Y^B ⊂ f(X^B). Hence Y^B = f(X^B). If in addition f is essential and Y ∈^⊥Pro(B), then Y =f(X^B) and hence X^B =X.

In view of Lemma 2.1, everyX ∈Pro(A) lies in a unique exact sequence

(2.1) 0−→X^B −→X −→XB −→0,

where X^B ∈ ^⊥Pro(B) and XB ∈ Pro(B). Moreover, every f ∈ HomPro(A)(X, Y) induces compatible morphisms

f^B :X^B −→Y^B, fB :XB −→YB. This defines a functor

π₀ =π₀^A,B : Pro(A)−→Pro(B), X 7−→XB. Since HomPro(A)(X^B, Y) = 0 for any Y ∈Pro(B), the natural map

HomPro(B)(XB, Y)−→HomPro(A)(X, Y)

is an isomorphism. In other words, π₀ is left adjoint to the inclusion of Pro(B) in Pro(A). As a consequence,π₀ is right exact and sends any projective object of Pro(A) to a projective object of Pro(B).

Lemma 2.2. — The functor π₀ commutes with filtered inverse limits.

Proof. — Consider a filtered inverse system (X_i) of objects of Pro(A). This yields a filtered inverse system (X_i^B) of objects of^⊥Pro(B); moreover, we have an isomorphism

lim→ HomPro(A)(X_i^B, Y)−→^∼= HomPro(A)(lim

← X_i^B, Y)

for any Y ∈ A (see [DG70, V.2.3.3]). Thus, Hom_Pro(A)(lim_←X_i^B, Y) = 0 for any Y ∈ B. In view of Lemma 2.1, it follows that lim←X_i^B ∈^⊥Pro(B). Also, we have an isomorphism

(lim← X_i)/(lim

← X_i^B)∼= lim

←(X_i)_B

by exactness of inverse limits (see [DG70, V.2.2]). So (lim←X_i)/(lim←X_i^B) is an

object of Pro(B); this yields the assertion.

We denote by

Q=Q^A,B : Pro(A)−→Pro(A)/Pro(B)

the quotient functor. Then Q is exact, and commutes with inverse limits in view of [Gab62, III.4.Prop. 9]. Also, recall from [Gab62, III.4.Prop. 8, Cor. 1] thatQ has a left adjoint: the cosection,

C =C^A,B : Pro(A)/Pro(B)−→Pro(A),

which also commutes with inverse limits and sends projectives to projectives. In other words, Pro(B) is a colocalizing subcategory of Pro(A), in the dual sense of [Gab62, III.2]. Conversely, every colocalizing subcategory of Pro(A) is equivalent to Pro(B) for a unique Serre subcategory B of A, in view of [Gab62, III.4.Prop. 10]

and [Bri19, Rem. 2.13]. Moreover, Pro(A)/Pro(B) is equivalent to Pro(A/B) by [Bri19, Prop. 2.12].

By [Gab62, III.2.Cor.], the essential image ofC consists of thoseX ∈Pro(A) such that

(2.2) HomPro(A)(X, Y) = 0 = Ext¹_Pro(A)(X, Y) for allY ∈Pro(B)

(these are the objects of the left perpendicular subcategory to Pro(B) in Pro(A), as defined in [GL91]). Moreover, for anyX ∈Pro(A), the adjunction mapCQ(X)→X has its kernel and cokernel in Pro(B) (see [GL91, III.2.Prop. 3]). This yields an exact sequence in Pro(A)

(2.3) 0−→Y₁ −→^ι X^f−→^ρ X −→^γ Y₀ −→0,

where we set X^f = X^fA,B := CQ(X) (in particular, X^f ∈ ^⊥Pro(B)), and we have Y₀, Y₁ ∈Pro(B). Note that the long exact sequence (2.3) depends functorially onX.

Also, note the natural isomorphism

HomPro(A)(X, Y^f )∼= HomPro(A)/Pro(B)(Q(X), Q(Y)) for any Y ∈Pro(A). In particular, if X, Y ∈ A then

(2.4) HomPro(A)(X, Y^f )∼= HomA/B(Q(X), Q(Y)).

Lemma 2.3. — With the above notation, we haveρ(X) =^f X^B and the induced epimorphism η:X^f→X^B is essential. Also, there are functorial isomorphisms

π0(X^B)−→^∼= Y0, Hom_Pro(B)(Y₁, Y)−→^∼= Ext¹_Pro(A)(X^B, Y) for all Y ∈Pro(B).

Proof. — In view of (2.2) and the exact sequence 0−→Y₁ −→X^f−→ρ(X)^f −→0,

we obtain the vanishing of Hom_Pro(A)(ρ(X), Y^f ) and an isomorphism HomPro(B)(Y₁, Y)−→^∼= Ext¹_Pro(A)(ρ(X), Y^f )

for allY ∈Pro(B). Thus, ρ(X)^f ∈^⊥Pro(B). SinceX/ρ(X)^f ∼=Y₀ ∈Pro(B), it follows that ρ(X) =^f X^B.

It remains to show that η:X^f→X^B is essential. Let Z be a subobject of X^fsuch that the compositionZ →X^f→X^B is an epimorphism. ThenX^f=Y₁+Z and hence X/Zf ∼= Y₁/(Y₁∩Z) ∈ Pro(B). As X^f∈ ^⊥Pro(B), it follows that Y₁/(Y₁ ∩Z) = 0,

i.e., Y₁ is a subobject of Z. Thus, Z =X.^f

Lemma 2.4. — With the notation of the exact sequence (2.3), the following conditions are equivalent for X ∈Pro(A):

(1) Y₀ =Y₁ = 0.

(2) Hom_Pro(A)(X, Y) = 0 = Ext¹_Pro(A)(X, Y)for all Y ∈Pro(B).

(3) HomPro(A)(X, Y) = 0 = Ext¹_Pro(A)(X, Y)for all Y ∈ B.

Proof. — The equivalence (1)⇔(2) holds by [Gab62, III.2.Cor.]. As (2)⇒(3) is obvious, it suffices to show that (3)⇒(1).

Let X ∈ Pro(A) satisfy (3), and Y ∈ Pro(B). Then HomPro(A)(X, Y) = 0 by Lemma 2.1 (1). Consider an essential epimorphism f : P → X, where P ∈ Pro(A) is projective (such a projective cover of X exists in view of [Gab62, II.6.Thm. 2]).

Then P ∈^⊥Pro(B) by Lemma 2.1 (3). So the exact sequence 0−→X⁰ −→P −→^f X −→0 yields isomorphisms

(2.5) Hom_Pro(A)(X⁰, Y)−→^∼= Ext¹_Pro(A)(X, Y)

for all Y ∈ Pro(B). In particular, Hom_Pro(A)(X⁰, Y) = 0 for all Y ∈ B. It follows that X⁰ ∈ ^⊥Pro(B) by using Lemma 2.1 (1). Thus, Ext¹_Pro(A)(X, Y) = 0 for all

Y ∈Pro(B).

2.2. Homotopy groups We denote by

π_i =π_i^A,B :=Lⁱπ₀^A,B : Pro(A)−→Pro(B) (i>0)

the left derived functors of the right exact functorπ₀. In view of Lemma 2.2 together with [DG70, V.2.3.8], the ith homotopy functor π_i commutes with filtered inverse limits for anyi>0. Also, for any exact sequence

0−→X₁ −→X −→X₂ −→0 in Pro(A), we have an associated homotopy exact sequence

(2.6) · · · →π_i+1(X₂)→π_i(X₁)→π_i(X)→π_i(X₂)→πi−1(X₁)→ · · ·

Lemma 2.5. — Assume that every projective object of Pro(B) is projective in Pro(A). Then:

(1) π_i(Y) = 0 for all Y ∈Pro(B) and i>1.

(2) πi(X^B)−→^∼= πi(X) for all X ∈Pro(A) and i>1.

Proof. —

(1). — Let P• be a projective resolution ofY in Pro(B). Then π₀(P•) = P• is still a projective resolution ofY in Pro(A).

(2). — This follows from (1) in view of the exact sequence (2.1).

Lemma 2.6. — With the assumption of Lemma 2.5, there is a functorial isomor- phismπ₁(X)∼=Y₁ for any X ∈Pro(A).

Proof. — The exact sequence (2.1) yields an isomorphism Q(X^B) → Q(X) in Pro(A)/Pro(B), and hence an isomorphismCQ(X^B)→CQ(X) in Pro(A). In turn, this yields an isomorphismY₁(X^B)→Y₁(X) in Pro(B), where Y₁(X^B) denotes the kernel of the adjunction map CQ(X^B)→X^B, andY₁(X) is defined similarly. Thus, we may assume that X ∈^⊥Pro(B). We then have an exact sequence

0−→Y₁ −→X^f−→X−→0, which yields an exact sequence

π₁(X)^f −→π₁(X)−→Y₁ −→π₀(X).^f

Moreover,π₀(X) = 0 by Lemma 2.4. So it suffices to show that^f π₁(X) = 0.^f As in the proof of Lemma 2.4, consider an exact sequence

0−→X⁰ −→P −→^f X^f−→0,

where P is projective and f is essential. We obtain an exact sequence π₁(P)−→π₁(X)^f −→π₀(X⁰)−→π₀(P).

Moreover, π₀(P) = 0 by Lemma 2.1 (3), and π₁(P) = 0 by definition. Thus, we have π₁(X)^f ∼=π₀(X⁰). Also, recall from (2.2) that Ext¹_Pro(A)(X, Y^f ) = 0 for allY ∈Pro(B).

Using the isomorphism (2.5), this yields HomPro(A)(X⁰, Y) = 0, and henceπ₀(X⁰) = 0.

Thus,π₁(X) = 0 as desired.^f

In view of Lemmas 2.3 and 2.6, the exact sequence (2.3) can be rewritten in a more suggestive way. Namely, with the assumption of Lemma 2.5, we have an exact sequence for anyX ∈Pro(A):

(2.7) 0−→π₁(X)−→^ιX X^f−→^ρX X −→^γX π₀(X)−→0.

In particular, whenX ∈^⊥Pro(B), we obtain an extension

(2.8) 0−→π1(X)−→X^f−→X −→0.

Using Lemmas 2.3 and 2.6 again, this yields in turn:

Corollary 2.7. — With the assumption of Lemma 2.5, let X ∈^⊥Pro(B)and Y ∈ Pro(B). Then HomPro(A)(π₁(X), Y) →^∼= Ext¹_Pro(A)(X, Y) via pushout by the extension (2.8).

In other words, (2.8) is the universal extension ofX by an object of Pro(B). We now record a similar uniqueness result for the exact sequence (2.7), to be used in Subsection 3.2.

Lemma 2.8. — With the assumption of Lemma 2.5, consider an exact sequence

(2.9) 0−→Y₁ −→X⁰ −→X −→Y₀ −→0

inPro(A), where Y₀, Y₁ ∈Pro(B) and X⁰ is in the essential image ofC. Then there is a commutative diagram of exact sequences

0 ^//π₁(X) ^//

f1

Xf ^//

f⁰

X ^//

f

π₀(X) ^//

f0

0

0 ^//Y₁ ^//X^{0 //}X ^//Y₀ ^// 0, where f₁, f⁰, f, f₀ are isomorphisms.

Proof. — Cut the exact sequence (2.9) in two short exact sequences 0−→Y₁ −→X⁰ −→X⁰⁰ −→0, 0−→X⁰⁰ −→X −→Y₀ −→0.

Since X⁰ is an object of ^⊥Pro(B), so is X⁰⁰. As Y₀ ∈ B, we obtain a commutative diagram of exact sequences

0 ^// X^{B //}

f⁰⁰

X ^//

f

π₀(X) ^//

f0

0

0 ^// X^{00 //}X ^//Y₀ ^// 0,

where the vertical arrows are isomorphisms. As a consequence, we may replace X with X^B, and assume that π₀(X) = 0 =Y₀.

Also, the induced morphism Q(X⁰) →Q(X) is an isomorphism, and hence so is CQ(X⁰) → CQ(X) = X. Since the adjunction^f CQ(X⁰) → X⁰ is an isomorphism, this yields an isomorphism X^f∼=X⁰. Thus, we may further assume that (2.9) is of the form

0−→Y1 −→X^f−→X−→0.

Then the associated map HomPro(B)(Y₁, Y)→Ext¹_Pro(A)(X, Y) is an isomorphism for all Y ∈Pro(B), by Lemma 2.4. In view of the uniqueness of the universal extension of X by an object of Pro(B), this completes the proof.

Next, we obtain two reformulations of the left exactness of the functorπ₁:

Lemma 2.9. — With the assumption of Lemma 2.5, the following conditions are equivalent:

(1) The cosection functor C : Pro(A)/Pro(B)→Pro(A)is exact.

(2) π₁ is left exact.

(3) π_i = 0 for all i>2.

Proof. —

(1)⇒(2). — Consider an exact sequence

0−→X₁ −→X −→X₂ −→0

in Pro(A). Then we have a commutative diagram of exact sequences 0 ^//X^f₁ ^//

Xf ^//

Xf₂ ^//

0

0 ^//X₁ ^//X ^//X₂ ^//0.

In view of the exact sequence (2.7) and its analogues forX1, X2, the snake lemma yields an exact sequence

0→π₁(X₁)→π₁(X)→π₁(X₂)→π₀(X₁)→π₀(X)→π₀(X₂)→0.

In particular,π₁ is left exact.

(2)⇒(1). — This follows from the dual statement of [Gab62, III.3.Prop. 7].

(2)⇒(3). — This is obtained by a standard argument that we recall for complete- ness. LetX ∈Pro(A) and choose a projective cover

0−→X⁰ −→P −→X −→0.

As πi(P) = 0 for all i>1, we obtain isomorphisms πi(X)→^∼= πi−1(X⁰) for all i>2.

SinceX⁰ is a subobject ofP, we have π₁(X⁰) = 0 by left exactness, henceπ₂(X) = 0.

Iterating this argument completes the proof.

(3)⇒(2). — This follows from the homotopy exact sequence (2.6).

Finally, we record an easy and useful divisibility property of homotopy groups. For anyX ∈Pro(A) and any integern, we denote bynX ∈EndA(X) the multiplication byn, and byX[n] its kernel. We say that X is divisible (resp. uniquely divisible) if n_X is an epimorphism (resp. an isomorphism) for any n >1.

Lemma 2.10. — With the assumption of Lemma 2.5, let X be an object of Pro(A). Assume thatX is divisible and X[n]∈Pro(B)for any n>1(in particular, π_i(X[n]) = 0 for any such n and any i > 1). Then X^f and the π_i(X) (i > 2) are uniquely divisible. Moreover, there is an exact sequence

0−→π₁(X)−→ⁿ π₁(X)−→X[n]−→π₀(X)−→ⁿ π₀(X)−→0 for any n>1.

Proof. — By assumption, we have an exact sequence

(2.10) 0−→X[n]−→X −→^nX X −→0

for anyn >1. Thus,n_X induces an automorphism ofQ(X), and hence of CQ(X) = X. In other words,f X^fis uniquely divisible. The remaining assertions follow from the

homotopy exact sequence associated with (2.10).

2.3. Structure of projective objects

In this subsection, we consider an artinian abelian category A and a Serre sub- category B such that every projective object of Pro(B) is projective in Pro(A).

Our aim is to describe the projectives of Pro(A) in terms of those of Pro(B) and Pro(A)/Pro(B)∼= Pro(A/B). We first obtain a generalization of [DG70, V.3.3.9]:

Lemma 2.11. — For any projective object P ∈Pro(A), there is an isomorphism P ∼=P^B⊕π₀(P) which is compatible withγ_P :P →π₀(P). Moreover, P^e ∼=P^B.

Proof. — Recall that π₀ is left adjoint to the inclusion of Pro(B) in Pro(A). It follows thatπ₀(P) is projective in Pro(B), and hence in Pro(A) as well. This yields a compatible isomorphismP ∼=P^B⊕π₀(P). In particular,P^B is projective, and hence in the essential image of C by (2.2). So the adjunction map CQ(P^B) → P^B is an isomorphism. AsCQ(P^B)→^∼= CQ(P) =P^e, this completes the proof.

Corollary 2.12. — Letf :X →Y be an epimorphism inPro(A), where Y is an object of Pro(B). Then there exists a subobject Y⁰ of X such that Y⁰ ∈ Pro(B) and the composition Y⁰ →X →Y is an epimorphism.

Proof. — We may assume that X is projective. By Lemma 2.11, we may then choose an isomorphism X ∼= X^f⊕π₀(X) compatibly with γ_X : X → π₀(X). Since π₀(f) :π₀(X)→π₀(Y) is an epimorphism, and γ_Y :Y →π₀(Y) is an isomorphism,

the statement holds withY⁰ =π₀(X).

The above corollary asserts that the pair (Pro(A),Pro(B)) satisfies the lifting property introduced in [Bri19, §2.2]. Thus, this property holds for the pair (A,B) as well. Conversely, if (A,B) satisfies the lifting property, then every projective object in Pro(B) is projective in Pro(A) by [Bri19, Lem. 2.14].

Next, recall from [DG70, V.2.4] that every projective object of Pro(A) is a product of indecomposable projectives, unique up to reordering; moreover, the indecompos- able projectives are projective covers of objects ofA. Also, givenX ∈Pro(A) such that Q(X) is projective in Pro(A/B), the adjunction map ρ : X^f = CQ(X) → X is the projective cover of X (indeed, C sends projectives to projectives, and ρ is essential by Lemma 2.3). Together with Lemma 2.11, this yields the following result (see also [Gab62, III.3.Cor. 2]):

Corollary 2.13. — The indecomposable projectives ofPro(A)are exactly those of Pro(B) and the X, where^f X ∈^⊥Pro(B) andQ(X) is indecomposable projective inPro(A/B).

The latter indecomposable projectives can be constructed as follows:

Lemma 2.14. — Let X ∈^⊥Pro(B).

(1) Consider an exact sequence in Pro(A),

0−→Z −→Y −→^f X −→0.

Then f is essential if and only ifZ ∈Pro(B) and Y ∈^⊥Pro(B).

(2) Assume that Q(X) is projective in Pro(A)/Pro(B). Then the essential epi- morphisms f : Y → X, where Ker(f) ∈ B, form a filtered inverse system with limit the projective cover ofX in Pro(A).

Proof. —

(1). — Note that f induces an epimorphism Y /Y^B → X/f(X^B). Since we have that Y /Y^B ∈ Pro(B) and X/f(X^B) ∈ ^⊥Pro(B), we must have X/f(X^B) = 0, i.e., the compositionY^B →Y →X is an epimorphism.

Assume thatf is essential. Then Y^B =Y, i.e., Y ∈^⊥Pro(B). Also, by the lifting property, we have Z =Z^B+W for some subobject W of Z such that W ∈Pro(B).

This yields an exact sequence

0−→Z/W −→Y /W −→X/f(W)−→0

in Pro(A), and hence in Pro(A/B). As X/f(W) ∼= X is projective in the latter category, this sequence is split by some g ∈ HomPro(A/B)(X, Y /W). In view of the assumption thatX ∈^⊥Pro(B), we may representg by h∈HomPro(A)(X, Y /W⁰) for someW⁰ ⊂ Y such that W ⊂W⁰ and W⁰ ∈Pro(B). Denote by p the composition of morphisms in Pro(A)

X −→^h Y /W⁰ −→X/f(W⁰)

(where the morphism on the right is induced by f), and by q : X → X/f(W⁰) the quotient morphism in Pro(A). Then p represents the identity endomorphism ofX in Pro(A)/Pro(B); thus, p−q represents zero there. Using again the assumption that X ∈ ^⊥Pro(B), it follows that p − q is zero in Pro(A). In particular, the composition h(X)→Y /W⁰ →X is an epimorphism. Since f is essential, hmust be an epimorphism as well. Sog is an isomorphism in Pro(A)/Pro(B), hence Z/W ∈ Pro(B). We conclude that Z ∈Pro(B).

Conversely, assume that Z ∈ Pro(B) and Y ∈ ^⊥Pro(B). Let Y⁰ ⊂ Y such that the composition Y⁰ → Y → X is an epimorphism. Then Y = Y⁰ +Z, hence Z → Y → Y /Y⁰ is an epimorphism as well. So Y /Y⁰ is an object of Pro(B), and hence is zero. We conclude thatf is essential.

(2). — Consider two exact sequences

0−→Z_i −→Y_i −→^fi X −→0 (i= 1,2),

where f₁, f₂ are essential and Z₁, Z₂ ∈ B. Then the induced morphism f :Y₁×_X Y₂ =:Y −→X

is an epimorphism with kernelZ₁×Z₂. In view of (1), it follows that the composition Y^B →Y →X is an essential epimorphism. Thus, these essential epimorphisms form a filtered inverse system.

Given such an essential epimorphism f : Y → X, the map ρ : X^f → X lifts to a morphism ϕ_Y : X^f → Y. Moreover, ϕ_Y is unique (since Ker(f) ∈ Pro(B) and Xf∈^⊥Pro(B)), and is an epimorphism as well. So we obtain an epimorphism

ϕ:X^f−→lim

← Y

with an obvious notation. To show that ϕ is a monomorphism, consider the family (K_i) of subobjects of Ker(ρ) such that Ker(ρ)/K_i ∈ B. Then X/K^f _i ∈ A and ρ factors through an essential epimorphismX/K^f _i →X; moreover, the corresponding morphism ϕ_i : X^f→ X/K^f _i is just the quotient morphism. Since ∩K_i is zero, this

completes the proof.

2.4. Compatibility properties

Throughout this subsection, we consider an artinian abelian categoryA, a Serre subcategoryBsuch that the pair (A,B) satisfies the lifting property, and in addition a Serre subcategory C of B. We first relate the homotopy functors associated to the three pairs (A,B), (B,C) and (A,C):

Lemma 2.15. — Let X ∈Pro(A).

(1) There is a natural isomorphism π₀^A,C(X)→^∼= π₀^B,C(π₀^A,B(X)).

(2) There is a spectral sequence π_i^B,C(π^A,B_j (X))⇒π^A,C_i+j(X).

Proof. —

(1). — This follows readily from the definitions.

(2). — Recall thatπ₀^A,B : Pro(A)→Pro(B) sends projectives to projectives; also, every projective in Pro(B) is obviously acyclic for π₀^B,C. In view of (1), this yields a

Grothendieck spectral sequence as stated.

Remark 2.16. — When X ∈ B, the above spectral sequence yields isomorphisms π^B,C_i (X)→^∼= π^A,C_i (X) for alli>0, in view of Lemma 2.5. Alternatively, these isomor- phisms follow from the obvious equality π₀^B,C(X) =π₀^A,C(X), since every projective object of Pro(B) is projective in Pro(A).

On the other hand, when X ∈^⊥Pro(B), the first terms of the spectral sequence yield a natural isomorphism

π^A,C₁ (X)−→^∼= π^B,C₀ (π₁^A,B(X)).

This can also be seen directly: consider the universal extension of X by an object of Pro(B),

0−→Y −→X^f−→X −→0,

where Y :=π₁^A,B(X). Then one may readily check that the induced exact sequence 0−→Y /Y^C −→X/Y^{f C} −→X −→0

is the universal extension ofX by an object of Pro(C); thus, Y /Y^C ∼=π₁^A,B(X). But also Y /Y^C =π₀^B,C(π₁^A,B(X)).

Next, we investigate the behavior of the homotopy groupsπ^A,B_i under the quotient functor

Q^A,C : Pro(A)−→Pro(A)/Pro(C).

We will need the following observation:

Lemma 2.17. — Assume that the pair (A,C) satisfies the lifting property. Then B/C is a Serre subcategory of A/C, and the quotient (A/C)/(B/C) is naturally equivalent to A/C. Moreover, the pair (A/C,B/C) satisfies the lifting property.

Proof. — Let X ∈ B, Y ∈ A, and let ϕ : X → Y be an isomorphism in A/C. By [Bri19, Lem. 2.7], there exists a subobject Y⁰ ⊂Y inA such that Y⁰ ∈ C and ϕ is represented by a morphism f :X →Y /Y⁰ in A. Then Ker(f) and Coker(f) are objects of C in view of [Gab62, III.1.Lem. 2]. Since B is a Serre subcategory of A containingC, it follows that Y ∈ B. Thus, B/C is a strict subcategory of A/C.

Next, let 0→X₁ →X →X₂ →0 be an exact sequence in A/C. Then there exists a commutative diagram in that category

0 ^//X₁ ^//

X ^//

X₂ ^//

0

0 ^//Y₁ ^//Y ^// Y₂ ^//0,

where the vertical arrows are isomorphisms, and the bottom sequence is the image of an exact sequence in A under the quotient functor Q^A,C (see [Bri19, Lem. 2.9]).

As a consequence, X ∈ B if and only if X₁, X₂ ∈ B. So B/C is a Serre subcategory of A/C.

The equivalence of categories (A/C)/(B/C) ∼= A/B follows from the universal property of quotient functors.

We now check that (A/C,B/C) satisfies the lifting property. Let ϕ : X → Y be an epimorphism in A/C. In view of [Bri19, Lem. 2.7] again, replacing Y with an isomorphic object in A/C, we may assume that ϕ is represented by a morphism f : X → Y in A; then Coker(f) is an object of C by [Gab62, III.1.Lem. 2] again.

Next, we may replaceX, Y withX^C, Y^C, and hence assume thatf is an epimorphism inA. Then there exists a subobject Y⁰ of X such thatY⁰ ∈ B and the composition

Y⁰ →X →Y is an epimorphism in A, hence in A/C.

Lemma 2.18. — With the assumption of Lemma 2.17, π_i^A/C,B/C(X) is naturally isomorphic to the image of π^A,B_i (X) inPro(B/C), for any i>0 and X ∈Pro(A).

Proof. — Recall that every projective object in Pro(C) is projective in Pro(A). By the dual statement of [Gab62, III.3.Cor. 3], it follows that the quotient functorQ^A,C sends projectives to projectives. Thus, it suffices to check the assertion for i= 0.

LetX ∈Pro(A) and consider the exact sequence

0−→X^B −→X −→π₀^A,B(X)−→0

in Pro(A), where X^B ∈^⊥Pro(B). This sequence is still exact in Pro(A/C); thus, it suffices to show thatX^B ∈^⊥Pro(B/C). In view of Lemma 2.1, it suffices in turn to show that every morphismϕ:X^B →Y in Pro(A/C), where Y ∈ B, is zero.

In Pro(A), we have X^B = lim←Xi, where Xi ∈ A and the projections X^B → Xi

are epimorphisms. Hence this also holds in Pro(A/C). Since Hom_Pro(A/C)(lim

← X_i, Y) = lim

→ Hom_A/C(X_i, Y),

we see that ϕ is represented by a morphism ϕ_i : X_i → Y in A/C. Using [Bri19, Lem. 2.7], it follows that ϕis represented by a morphism fi :Xi →Y /Y⁰ inA, for someY⁰ ⊂Y such that Y⁰ ∈ C. The composition X^B → Xi → Y /Y⁰ is zero, since Y /Y⁰ ∈ B. So f_i = 0, and ϕ= 0.

3. Fundamental groups of commutative algebraic groups

3.1. The affine fundamental group

Let k be a field. As in the introduction, we consider the artinian abelian category C of commutative k-group schemes of finite type, and the associated pro-category Pro(C) of pro-algebraic groups. We denote byLthe full subcategory ofC with objects the affine (or equivalently, linear) algebraic groups. ThenLis a Serre subcategory of C, as follows from fpqc descent (see e.g. [Sta18, 34.20.18]). Also, recall that Pro(L) is equivalent to the category of commutative affine k-group schemes.

By the results of Subsection 2.1, every object of Pro(C) has a largest affine quotient;

this yields a right exact functor

π₀^C,L: Pro(C)−→Pro(L),

which commutes with filtered inverse limits and extends the affinization functor C → Lconsidered for example in [DG70, III.3.8]. The results of Subsection 2.2 also apply to this setting, in view of the following observation:

Lemma 3.1. — The pair (C,L) satisfies the lifting property.

Proof. — Let G∈ C. By a variant of Chevalley’s structure theorem for algebraic groups (see [Bri17, Thm. 2.3]) that we will use repeatedly, there is an exact sequence

(3.1) 0−→L−→G−→A−→0,

where Lis a linear algebraic group andAis an abelian variety. Letf :G→H be an epimorphism, whereHis linear. ThenG⁰ :=G/(Ker(f)+L) is linear (as a quotient of G/Ker(f)∼=H) and is an abelian variety (as a quotient of G/L∼=A). Thus,G⁰ = 0, i.e., G= Ker(f) +L. So the composition L→G→H is an epimorphism.

We now describe the quotient categories C/L and Pro(C)/Pro(L). Consider the full subcategory A of C with objects the abelian varieties; then A is an addi- tive subcategory, but not a Serre subcategory. Denote by A the corresponding isogeny category: the objects of A are those of A, and the morphisms are defined by HomA(G, H) := HomA(G, H)⊗_Z Q. Then A is a semi-simple artinian abelian category; its simple objects are exactly the simple abelian varieties, i.e., those having no non-trivial abelian subvariety.

Lemma 3.2. — With the above notation, the composite functor A → C → C/L induces equivalences of categories

A−→ C^∼= /L, Pro(A)−→^∼= Pro(C)/Pro(L).

Moreover, Pro(A) is semi-simple.

Proof. — Denote by F : A → C/L the composite functor. Then F is essentially surjective by Chevalley’s theorem again. Also, recall from [Gab62, III.1] that

Hom_C/L(G, H) = lim

→ HomC(G⁰, H/H⁰)

for allG, H ∈ C, whereG⁰ (resp.H⁰) runs over the subgroup schemes ofGsuch that G/G⁰ is linear (resp. the linear subgroup schemes ofH). WhenGand H are abelian varieties, we must have G⁰ =G; moreover,H⁰ is finite, or equivalently, contained in the n-torsion subgroup scheme H[n] for some n>1. As a consequence,

HomC/L(G, H) = lim

→ HomC(G, H/H[n]),

where the direct limit is over the positive integers ordered by divisibility. This yields a natural isomorphism

Hom_C/L(G, H)−→^∼= Hom_C(G, H)⊗_ZQ

(see e.g. [Bri17, Prop. 3.6] for details), and hence the first equivalence of categories, A ∼=C/L. Since Pro(C/L)∼= Pro(C)/Pro(L), this yields the second equivalence of categories Pro(A)∼= Pro(C)/Pro(L).

To show that Pro(A) is semi-simple, it suffices to check that every object is projective. In view of [DG70, V.2.3.5], it suffices in turn to check that for any object G of Pro(A) and any epimorphism f : G₁ → G₂ in A, the induced map HomA(G, G₁)→HomA(G, G₂) is surjective. But this follows from the existence of a

section off.

Before stating our next result, we introduce some notation. We denote by Q=Q^C,L: Pro(C)−→Pro(C)/Pro(L)

the quotient functor, and by

C =C^C,L: Pro(C)/Pro(L)−→Pro(C) the associated cosection functor. For any abelian varietyA, we set

P(A) :=CQ(A).

Proposition 3.3. — (1) The functor C is exact.

(2) The projective objects of Pro(C)are exactly the products of those of Pro(L) with theP(A), where Ais an abelian variety. Moreover, P(A)is a projective cover ofA in Pro(C), and is uniquely divisible.

Proof. —

(1). — Recall thatCcommutes with inverse limits, and hence with products. Since the category Pro(C)/Pro(L) is semi-simple (Lemma 3.2), this yields the assertion.

(2). — By Lemma 3.1 and [Bri19, Lem. 2.14], every projective object of Pro(L) is projective in Pro(C). In view of the dual statement of [Gab62, III.3.Cor. 2], it follows that the projective objects of Pro(C) are exactly the products of those of Pro(L) with the images under C of projective objects of Pro(C)/Pro(L). Using again the fact that this quotient category is semi-simple, this yields the first assertion.

LetAbe an abelian variety. Since every affine quotient ofAis trivial, the adjunction mapρ:P(A)→Ais an epimorphism. Also, ρis essential by Lemma 2.3; thus,P(A) is a projective cover of A in Pro(C). The unique divisibility assertion follows from Lemma 2.10, sinceAis divisible and itsn-torsion subgroup schemes are finite for all

n>1.

3.2. The profinite fundamental group

We now consider the Serre subcategory F of L with objects the finite group schemes. As in the introduction, we denote by

$_i :=π^C,F_i : Pro(C)−→Pro(F)

the profinite homotopy functors. For anyG∈Pro(C), the exact sequence (2.7) may be rewritten as

0−→$₁(G)−→G^e −→G−→$₀(G)−→0,

where G^e denotes the profinite universal cover of G^F := Ker(G→$0(G)).

The pair (C,F) satisfies the lifting property in view of [Bri15, Thm. 1.1]; thus, we may again use the constructions and results of Section 2.

Lemma 3.4. — Let G∈Pro(C) be divisible.

(1) G[n]is profinite for any n >1.

(2) $₀(G) = 0.

(3) G^e is the limit of the filtered inverse system (G, n_G)_n>1, where the positive integers are ordered by divisibility. Also, G^e is uniquely divisible.

(4) $₁(G) = lim←G[n], where the limit is over the above system. Moreover, we have $₁(G)/n$₁(G)∼=G[n] for any n >1.

(5) $_i(G) = 0 for any i>2.

Proof. —

(1). — Let G = lim_←Gi, where the Gi are algebraic groups and the projections G→G_i are epimorphisms. Then the induced map G[n]→lim_←G_i[n] is a monomor- phism. Moreover, each G_i is divisible (as a quotient of G); thus, G_i[n] is finite for dimension reasons. So lim←G_i[n] is profinite.

(2). — Consider an epimorphism G→H, where H ∈ F. Then H is divisible (as a quotient of G) and torsion (as a finite group scheme), hence zero. This yields the assertion.

(3). — Let G⁰ := lim←G (limit over the above system). For anyH ∈ C and i>0, we have

Extⁱ_Pro(C)(G⁰, H)∼= lim_→ Extⁱ_Pro(C)(G, H)

in view of [DG70, V.2.3.9]. Assume that H ∈ F; then we may choose an integer n>1 such that n_H = 0. Thus, Extⁱ_Pro(C)(G, H) is killed byn; as a consequence, we have that Extⁱ_Pro(C)(G⁰, H) = 0. Using Lemma 2.4, it follows that the adjunction map CQ(G⁰)→G⁰ is an isomorphism.

The projection π:G⁰ →G associated withn = 1, lies in an exact sequence

(3.2) 0−→lim

← G[n]−→G⁰ −→^π G−→0,

where lim←G[n] is profinite. Thus, π induces an isomorphism CQ(G⁰)→CQ(G) = G. So we may identifye G⁰ withG. Then (3.2) is identified with the universal profinite^e extension ofG, in view of Lemma 2.8.

(4). — The first assertion has just been proved; the second one follows from Lemma 2.10 in view of the vanishing of$₀(G).

(5). — By Lemma 2.10 again, the profinite group scheme$_i(G) is uniquely divisi- ble for anyi>2. As a consequence, every finite quotient of$_i(G) is divisible, hence

zero. This yields the assertion.

We may now prove a large part of our main result:

Theorem 3.5. — Assume that k is perfect.

(1) We have $_i = 0 for all i>2; equivalently,$₁ is left exact.

(2) The cosection functor C : Pro(C)/Pro(F)→Pro(C) is exact.

(3) The profinite universal cover G^e has projective dimension at most 1, for any G∈Pro(C).

Proof. —

(1). — In view of the homotopy exact sequence and the fact that $_i commutes with filtered inverse limits, it suffices to show that$_i(G) = 0 for any G∈ C and any i>2. This follows from Lemma 3.4 when Gis an abelian variety. On the other hand, when G ∈ L, we have $_i(G) = π^L,F_i (G) in view of Remark 2.16 and Lemma 3.1.

So the assertion follows from [DG70, V.3.6.8] in that case. In the general case, just recall that every G ∈ C is an extension of an abelian variety by a linear algebraic group.

(2). — This is just a reformulation of (1) (see Lemma 2.9).

(3). — By the main result of [Bri17], the category C/F has homological dimen- sion 1; hence the same holds for the category Pro(C)/Pro(F) ∼= Pro(C/F) (see e.g. [Bri19, Prop. 2.12, Lem. 2.15]). AsC sends projectives to projectives, this yields

the assertion.

Remark 3.6. — Returning to an arbitrary ground fieldk, consider the full subcate- goryE ofC with objects the finite étale group schemes. ThenE is a Serre subcategory ofF; moreover, the pair (C,E) satisfies the lifting property if and only if k is perfect (see e.g. [Bri15, Thm. 1.1, Rem. 3.3]). The functors

π_i :=π_i^C,E : Pro(C)−→Pro(E)

are the “pro-étale homotopy functors”, considered in [DG70, V.3.4.1] for affine group schemes over perfect fields; note that π0(G) = G/G⁰ for any G ∈ C, where G⁰ denotes the neutral component (see e.g. [DG70, II.5.1]). The functor

π^F,E₀ : Pro(F)→Pro(E)

is exact in view of [DG70, V.3.1.5]; using Lemma 2.15, this yields natural isomor- phisms

π_i(G)−→^∼= π^F₀^,E(π^C,F_i (G))

for allG∈Pro(C) and all i>0. As a consequence, the pro-étale fundamental group π₁ is left exact when k is perfect.

3.3. Projective covers of abelian varieties

Consider an abelian varietyA, and its projective coverP(A) in Pro(C). By Propo- sition 3.3, we have an exact sequence in Pro(C)

(3.3) 0−→L(A)−→P(A)−→^ρ A−→0,

where L(A) is affine. Also, recall that (3.3) is the universal affine extension of A, that is, the pushout by this extension yields an isomorphism

(3.4) HomPro(L)(L(A), G)−→^∼= Ext¹_Pro(C)(A, G) for any G∈Pro(L).

Next, note that an algebraic group G is an object of ^⊥Pro(L) if and only if G is anti-affine, i.e., O(G) = k (as follows from the affinization theorem, see [DG70, III.3.8.2]). In view of Lemma 2.14, it follows that P(A) is the inverse limit of all anti-affine extensions ofA. Using the affinization theorem again, one can deduce that the exact sequence (3.3) is the universal affine extension ofA by a (not necessarily commutative) affinek-group scheme. One can also obtain a structure result forP(A) by using the classification of anti-affine groups (see [Bri09, Thm. 2.7]). We will rather obtain such a result (Theorem 3.10) via an alternative approach, which relatesP(A) to the universal profinite cover ofA.

Consider the exact sequence as in (2.1),

0−→L(A)^F −→L(A)−→$₀(L(A))−→0.

Then the induced exact sequence

0−→$₀(L(A))−→P(A)/L(A)^F −→A−→0

is the universal profinite extension ofA, as observed in Remark 2.16. We thus identify

$₀(L(A)) with $₁(A), and P(A)/L(A)^F with the profinite universal cover A. This^e yields an exact sequence

(3.5) 0−→L(A)^F −→P(A)−→A^e −→0.

Lemma 3.7. — With the above notation, $_i(A) = 0 =^e $_i(L(A)^F) for anyi>0.

Proof. — Since A is divisible, we have $_i(A) = 0 for i>2 in view of Lemma 3.4.

Using the homotopy exact sequence associated with the universal profinite extension 0−→$1(A)−→A^e −→A−→0

together with Lemma 2.5, it follows that $_i(A) = 0 for^e i>2 as well. Also, we have by construction $₀(A) = 0 =^e $₁(A).^e

The assertion on the $_i(L(A)^F) follows by using the exact sequence (3.5).